A note on q-nanlogue of Sandor's functions

Scientia Magna Vol. 2 (2006), No. 1, 30-34

A note on q-nanlogue of S´andor’s functions Taekyun Kim Department of Mathematics Education Kongju National University Kongju 314- 701 , South Korea C. Adiga and Jung Hun Han Department of Studies in Mathematics University of Mysore Manasagangotri Mysore 570006, India Abstract The additive analogues of Pseudo-Smarandache, Smarandache-simple functions and their duals have been recently studied by J. S´ andor. In this note, we obtain q-analogues of S´ andor’s theorems [6].

Keywords q-gamma function; Pseudo-Smarandache function; Smarandache-simple function; Asymtotic formula.

Dedicated to Sun-Yi Park on 90th birthday

§1. Introduction The additive analogues of Smarandache functions S and S∗ have been introduced by S´ andor [5] as follows: S(x) = min{m ∈ N : x ≤ m!},

x ∈ (1, ∞),

S∗ (x) = max{m ∈ N : m! ≤ x},

x ∈ [1, ∞),

and

He has studied many important properties of S∗ relating to continuity, differentiability and Riemann integrability and also p roved the following theorems: Theorem 1.1.

S∗ ∼ Theorem 1.2.

The series

log x log log x ∞ X

(x → ∞).

1 , n(S∗ (n))α n=1 is convergent for α > 1 and divergent for α ≤ 1.

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In [1], Adiga and Kim have obtained generalizations of Theorems 1.1 and 1.2 by the use of Euler’s gamma function. Recently Adiga-Kim-Somashekara-Fathima [2] have established a qanalogues of these results on employing analogues of Pseudo-Smarandache, Smarandache-simple functions and their duals as follows: ½ ¾ m(m + 1) Z(x) = min m ∈ N : x ≤ , 2

x ∈ (0, ∞),

½ ¾ m(m + 1) Z∗ (x) = max m ∈ N : ≤x , 2

x ∈ [1, ∞),

P (x) = min{m ∈ N : px ≤ m!},

p > 1,

P∗ (x) = max{m ∈ N : m! ≤ px }, He has also proved the following theorems: Theorem 1.3.

1√ Z∗ ∼ 8x + 1 2 Theorem 1.4. The series ∞ X 1 n=1

x ∈ (0, ∞),

p > 1,

x ∈ [1, ∞).

(x → ∞).

(Z∗ (n))α

,

is convergent for α > 2 and divergent for α ≤ 2. The series ∞ X

1 , n(Z∗ (n))α n=1 is convergent for all α > 0.

Theorem 1.5. log P∗ (x) ∼ log x (x → ∞), Theorem 1.6.

The series

µ ¶α ∞ X 1 log log n n log P∗ (n) n=1

is convergent for all α > 1 and divergent for α ≤ 1.

The main purpose of this note is to obtain q-analogues of S´ andor’s Theorems 1.3 and 1.5.

In what follows, we make u se of the following notations and definitions. F. H. Jackson defined a q-analogues of the gamma function which extends the q-factorial (n!)q = 1(1 + q)(1 + q + q 2 ) · · · (1 + q + q 2 + · · · + q n−1 ),

cf

[3],

which becomes the ordinary factorial as q → 1. He defined the q-analogue of the gamma function as Γq (x) =

(q; q)∞ (1 − q)1−x , (q x ; q)∞

0 < q < 1,

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No. 1

and Γq (x) =

x (q −1 ; q −1 )∞ (1 − q)1−x q (2) , q > 1, (q −x ; q −1 )∞

where (a; q)∞ =

∞ Y

(1 − aq n ).

n=0

It is well known that Γq (x) → Γ(x) as q → 1, where Γ(x) is the ordinary gamma function.

§2. Main Theorems We now defined the q-analogues of Z and Z∗ as follows: ½ ¾ 1 − qm Γq (m + 2) Zq (x) = min :x≤ , m ∈ N, 1−q 2Γq (m) and

½ Zq∗ (x)

= max

¾ 1 − q m Γq (m + 2) : ≤x , 1−q 2Γq (m)

x ∈ (0, ∞),

·

m ∈ N,

¶ Γq (m + 2) x∈ ,∞ , 2Γq (1)

where 0 < q < 1. Clearly, Zq (x) → Z(x) and Zq∗ (x) → Z∗ (x) as q → 1− . From the definitions of Zq and Zq∗ , it is clear that  ³ i  Γ (3)  1, if x ∈ 0, 2Γqq (1)  ³ i Zq (x) = (1)  1−qm , ifx ∈ Γq (m+1) , Γq (m+2) , m ≥ 2, 1−q 2Γq (m−1) 2Γq (m) and Zq∗ =

1 − qm 1−q

· if x ∈

Γq (m + 2) Γq (m + 3) , 2Γq (m) 2Γq (m + 1)

¶ .

(2)

Since 1 − q m−1 1 − qm 1 − q m−1 1 − q m−1 ≤ = + q m−1 ≤ + 1, 1−q 1−q 1−q 1−q (1) and (2) imply that for x >

Γq (3) 2Γq (1) ,

Zq∗ ≤ Zq ≤ Zq∗ + 1.

Hence it suffices to study the function Zq∗ . We now prove our main theorems.

Theorem 2.1. If 0 < q < 1, then √ √ 1 + 8xq − (1 + 2q) 1 + 8xq − 1 Γq (3) ∗ < Zq ≤ , x≥ .

2 2q 2q 2Γq (1) Proof. If Γq (k + 2) Γq (k + 3) ≤x< , 2Γq (k) 2Γq (k + 1)

(3)

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A note on q-nanlogue of S´ andor’s functions

then Zq∗ =

1 − qk 1−q

and (1 − q k )(1 − q k+1 ) − 2x(1 − q)2 ≤ 0 < (1 − q k+1 )(1 − q k+2 ) − 2x(1 − q)2 .

(4)

Consider the functions f and g defined by f (y) = (1 − y)(1 − yq) − 2x(1 − q)2 and g(y) = (1 − yq)(1 − yq 2 ) − 2x(1 − q)2 .

Note that f is monotonically decreasing for y ≤ Also f (y1 ) = 0 = g(y2 ) where

1+q 2q

and g is strictly decreasing for y ≤

1+q 2q 2 .

√ (1 + q) − (1 − q) 1 + 8xq , 2q √ (q + q 2 ) − q(1 − q) 1 + 8xq . y2 = 2q 3 y1 =

Since y1 ≤

1+q 2q ,

y2 ≤

1+q 2q 2

and q k <

1+q 2q

<

1+q 2q 2 ,

from (4), it follows that

f (q k ) ≤ f (y1 ) = 0 = g(y2 ) < g(q k ).

Thus y1 < q k < y2 and hence 1 − qk 1 − y1 1 − y2 < < .

1−q 1−q 1−q i.e.

√

1 + 8xq − (1 + 2q) < Zq∗ ≤ 2q 2

√

1 + 8xq − 1 .

2q

This completes the proof.

Remark. Letting q − 1− in the above theorem, we obtain S´ andor’s Theorem 1.3.

We define the q-analogues of P and P∗ as follows: Pq (x) = min{m ∈ N : px ≤ Γq (m + 1)},

p > 1,

x ∈ (0, ∞),

Pq∗ (x) = max{m ∈ N : Γq (m + 1) ≤ px },

p > 1,

x ∈ [1, ∞),

and where 0 < q < 1. Clearly, Pq (x) → P (x) and Pq∗ → P∗ (x) as q → 1− . From the definitions of Pq and Pq∗ , we have Pq∗ (x) ≤ Pq (x) ≤ Pq∗ (x) + 1.

Hence it is enough to study the function Pq∗ .

Theorem 2.2. If 0 < q < 1, then P∗ (x) ∼

x log p ´ ³ 1 log 1−q

(x → ∞).

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No. 1

Proof. If Γq (n + 1) ≤ px < Γq (n + 2), then Pq∗ (x) = n and log Γq (n + 1) ≤ log px < log Γq (n + 2).

But by the q-analogue of Stirling’s formula established by Moak [4], we have µ ¶ µ n+1 ¶ µ ¶ 1 q 1 log Γq (n + 1) ∼ n + log ∼ n log .

2 q−1 1−q ³ ´ 1 Dividing (5) throughout by n log 1−q , we obtain log Γq (n + 1) x log p log Γq (n + 2) ³ ´ ≤ ´< ³ ´. ³ 1 1 1 n log 1−q Pq∗ (x) log 1−q n log 1−q

(5)

(6)

(7)

Using (6) and (7), we deduce lim

x→∞

x log p ³ Pq∗ (x) log

1 1−q

´ = 1.

This completes the proof.

References [1] C. Adiga and T. Kim, On a generalization of S´ andor’s function, Proc. Jangjeon Math.

Soc., 5(2002), 121-124.

[2] C. Adiga, T. Kim, D. D. Somashekara and N. Fathima, On a q-analogue of S´ andor’s function, J. Inequal. Pure and Appl. Math., 4(2003), 1-5.

[3] T. Kim, Non-archimedean q-integrals with multiple Changhee q-Bernoulli polynomials, Russian J. Math. Phys., 10(2003), 91-98.

[4] D. S. Moak, The q-analogue of Stirling’s formula, Rocky Mountain J. Math., 14(1984), 403-413.

[5] J. S´ andor, On an additive analogue of the function S, Note Numb. Th. Discr. Math., 7(2001), 91-95.

[6] J. S´ andor, On an additive analogue of certain arithmetic function, J. Smarandache Notions, 14(2004), 128-133.